Question

The value of $\frac{\cos \left(30°\right)+i\sin \left(30°\right)}{\cos \left(30°\right)-i\sin \left(30°\right)}$is equal to

• A. $i$
• B. $-i$
• C. $\frac{1+i\sqrt{3}}{2}$
• D. $\frac{1-i\sqrt{3}}{2}$
• E. $1+i$

Answer 1

The correct option is A

$i$

Explanation for correct option

Given trigonometric function is $\frac{\cos \left(30°\right)+i\sin \left(30°\right)}{\cos \left(60°\right)-i\sin \left(60°\right)}$

$$\begin{gathered} \frac{\cos \left(30°\right)+i\sin \left(30°\right)}{\cos \left(60°\right)-i\sin \left(60°\right)} \\ =\frac{{\displaystyle \frac{\sqrt{3}}{2}}+i{\displaystyle \frac{1}{2}}}{\frac{1}{2}-i\frac{\sqrt{3}}{2}} \\ =\frac{\sqrt{3}+i}{1-i\sqrt{3}} \\ =\frac{\left(\sqrt{3}+i\right)\left(1+i\sqrt{3}\right)}{\left(1-i\sqrt{3}\right)\left(1+i\sqrt{3}\right)} \\ =\frac{\sqrt{3}+i-\sqrt{3}+3i}{1+3} \\ =\frac{4i}{4} \\ =i \end{gathered}$$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{A}\right)$